Two-particle correlations in a dynamic cluster approximation with continuous momentum dependence: Superconductivity in the 2D Hubbard model

The DCA$^+$ algortihm was recently introduced to extend the dynamic cluster approximation (DCA) with a continuous lattice self-energy in order to achieve better convergence with cluster size. Here we extend the DCA$^+$ algorithm to the calculation of two-particle correlation functions by introducing irreducible vertex functions with continuous momentum dependence consistent with the DCA$^+$ self-energy. This enables a significantly more controlled and reliable study of phase transitions than with the DCA. We test the new method by calculating the superconducting transition temperature $T_{c}$ in the attractive Hubbard model and show that it reproduces previous high-precision determinantal quantum Monte Carlo results. We then calculate $T_c$ in the doped repulsive Hubbard model, for which previous DCA calculations could only access the weak-coupling ($U=4t$) regime for large clusters. We show that the new algorithm provides access to much larger clusters and delivers asymptotically converged results for $T_c$ for both the weak ($U=4t$) and intermediate ($U=7t$) coupling regimes, and thereby enables the accurate determination of the exact infinite cluster size result.